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It is clear from this diagram that the components of the stress, viewed from the x", x 2 , x-coordinate system, are not in Xi x• 1 x1 Fig. 9 A simple example of a pure shear in the xj frame which transforms into a pure tension and compression in the x' frame. shear but in tension and compression. It is because of this fact that E, v, and G are not independent constants. , viewed in the ax-frame serves to illustrate this point. 2 or Appendix G), in which ai, = 0 . 28) that the material undergoes the strain (c) el13 = The advantage of writing the stress and strain as tensors is that their components can be found in the x'-frame by means of the transformations T1i = a=ait T, and ei = aikaJlekj.

As the frequency is increased, the longitudinal wavelengths take on the same magnitude as the transverse dimensions of the elastic structure. 3. 4, the effect of boundaries is usually to couple shearing and dilatational motions of the material. As a result, the higher order modes, which become significant as the frequency is raised, are often mathematically complicated. 18 shows a slab of elastic material with a thickness d. We x3 Fig. 18 Slab of elastic material with thickness d and extending to infinity in the zsdirection.

V = Os in ,t •I - (b) This solution can be justified by direct substitution into (a) and can be thought of as a wave propagating with the phase velocity a. in the x-direction. Within an arbitrary constant that would be determined by the boundary conditions, the actual displacements follow from y = V. S. 6 =Y-a, cos (c) - At a given instant these displacements appear as shown in Fig. 3. Note that the material is displaced out of the regions of positive i and into regions of negative v'. We can imagine painting equidistant parallel lines in the unstressed material.